We use a strong-disorder renormalization group (SDRG) method and ground-state quantum Monte Carlo (QMC) simulations to study S = 1/2 spin chains with random couplings, calculating disorder-averaged spin and dimer correlations. The QMC simulations demonstrate logarithmic corrections to the power-law decaying correlations obtained with the SDRG scheme. The same asymptotic forms apply both for systems with standard Heisenberg exchange and for certain multispin couplings leading to spontaneous dimerization in the clean system. We show that the logarithmic corrections arise in the valence-bond (singlet pair) basis from a contribution that can not be generated by the SDRG scheme. In the model with multi-spin couplings, where the clean system dimerizes spontaneously, random singlets form between spinons localized at domain walls in the presence of disorder. This amorphous valence-bond solid is asymptotically a random-singlet state and only differs from the random-exchange Heisenberg chain in its short-distance properties.

We study dynamical properties at finite temperature (T ) of Heisenberg spin chains with random antiferrormagnetic exchange couplings, which realize the random singlet phase in the low-energy limit, using three complementary numerical methods: exact diagonalization, matrix-product-state algorithms, and stochastic analytic continuation of quantum Monte Carlo results in imaginary time. Specifically, we investigate the dynamic spin structure factor S(q, ω) and its ω → 0 limit, which are closely related to inelastic neutron scattering and nuclear magnetic resonance (NMR) experiments (through the spin-lattice relaxation rate 1/T1). Our study reveals a continuous narrow band of low-energy excitations in S(q, ω), extending throughout the q-space, instead of being restricted to q ≈ 0 and q ≈ π as found in the uniform system. Close to q = π, the scaling properties of these excitations are well captured by the random-singlet theory, but disagreements also exist with some aspects of the predicted q-dependence further away from q = π. Furthermore we also find spin diffusion effects close to q = 0 that are not contained within the random-singlet theory but give non-negligible contributions to the mean 1/T1. To compare with NMR experiments, we consider the distribution of the local relaxation rates 1/T1. We show that the local 1/T1 values are broadly distributed, approximately according to a stretched exponential. The mean 1/T1 first decreases with T , but below a crossover temperature it starts to increase and likely diverges in the limit of a small nuclear resonance frequency ω0. Although a similar divergent behavior has been predicted and experimentally observed for the static uniform susceptibility, this divergent behavior of the mean 1/T1 has never been experimentally observed. Indeed, we show that the divergence of the mean 1/T1 is due to rare events in the disordered chains and is concealed in experiments, where the typical 1/T1 value is accessed. *

We investigate the short time quantum critical dynamics in the imaginary time relaxation processes of finite size systems. Universal scaling behaviors exist in the imaginary time evolution and in particular, the system undergoes a critical initial slip stage characterized by an exponent θ, in which an initial power-law increase emerges in the imaginary time correlation function when the initial state has zero order parameter and vanishing correlation length. Under different initial conditions, the quantum critical point and critical exponents can be determined from the universal scaling behaviors. We apply the method to the one-and two-dimensional transverse field Ising models using quantum Monte Carlo simulations. In the one-dimensional case, we locate the quantum critical point at (h/J)c = 1.00003(8) in the thermodynamic limit, and estimate the critical initial slip exponent θ = 0.3734(2), static exponent β/ν = 0.1251(2) by analyzing data on chains of length L = 32 ∼ 256 and L = 48 ∼ 256, respectively. For the two-dimensional square-lattice system, the critical coupling ratio is given by 3.04451(7) in the thermodynamic limit while the critical exponents are θ = 0.209(4) and β/ν = 0.518(1) estimated by data on systems of size L = 24 ∼ 64 and L = 32 ∼ 64, correspondingly. Remarkably, the critical initial slip exponents obtained in both models are notably distinct from their classical counterparts, owing to the essential differences between classical and quantum dynamics. The short time critical dynamics and the imaginary time relaxation QMC approach can be readily adapted to various models.

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